## Abstract

An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper continues the investigation on integrals of groups started in the work [1]. We study:

-- A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2).

-- The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1).

-- Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5).

-- The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7).

-- Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1).

-- Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2).

We end the paper with a number of open problems.

-- A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2).

-- The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1).

-- Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5).

-- The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7).

-- Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1).

-- Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2).

We end the paper with a number of open problems.

Original language | English |
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Number of pages | 43 |

Journal | Israel Journal of Mathematics |

Early online date | 24 Apr 2024 |

DOIs | |

Publication status | E-pub ahead of print - 24 Apr 2024 |